Scattering and absorption property database for
nonspherical ice particles in the near- through
far-infrared spectral region
Ping Yang, Heli Wei, Hung-Lung Huang, Bryan A. Baum, Yong X. Hu,
George W. Kattawar, Michael I. Mishchenko, and Qiang Fu
The single-scattering properties of ice particles in the near- through far-infrared spectral region are
computed from a composite method that is based on a combination of the finite-difference time-domain
technique, the T-matrix method, an improved geometrical-optics method, and Lorenz–Mie theory. Seven
nonspherical ice crystal habits (aggregates, hexagonal solid and hollow columns, hexagonal plates, bullet
rosettes, spheroids, and droxtals) are considered. A database of the single-scattering properties for each
of these ice particles has been developed at 49 wavelengths between 3 and 100 !m and for particle sizes
ranging from 2 to 10,000 !m specified in terms of the particle maximum dimension. The spectral
variations of the single-scattering properties are discussed, as well as their dependence on the particle
maximum dimension and effective particle size. The comparisons show that the assumption of spherical
ice particles in the near-IR through far-IR region is generally not optimal for radiative transfer computation. Furthermore, a parameterization of the bulk optical properties is developed for mid-latitude cirrus
clouds based on a set of 21 particle size distributions obtained from various field campaigns. © 2005
Optical Society of America
OCIS codes: 010.1310, 010.1290, 290.5850, 290.1090, 300.6170, 260.3060.
1. Introduction
The radiative properties of ice clouds have been studied from various perspectives.1– 8 In particular, substantial research efforts have been focused on the
optical properties of ice particles.9 –21 In the past decade, numerous techniques were developed to use
high-spectral-resolution IR interferometer measurements for the determination of cloud properties, e.g.,
that developed by Smith et al.22 Most of these retrieval methods consider measurements primarily
P. Yang (pyang@ariel.met.tamu.edu) and H. Wei are with the
Department of Atmospheric Sciences, Texas A&M University, College Station, Texas 77843. H.-L. Huang is with the Cooperative
Institute for Meteorological Satellite Studies, University of Wisconsin, Madison, Wisconsin 53706. B. A. Baum and Y. X. Hu are
with NASA Langley Research Center, Hampton, Virginia
23681. G. W. Kattawar is with the Department of Physics, Texas
A&M University, College Station, Texas 77843. M. I. Mishchenko
is with NASA Goddard Institute for Space Studies, New York, New
York 10025. Q. Fu is with the Department of Atmospheric Science,
University of Washington, Seattle, Washington 98195.
Received 20 October 2004; revised manuscript received 13 February 2005; accepted 5 April 2005.
0003-6935/05/265512-12$15.00/0
© 2005 Optical Society of America
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APPLIED OPTICS ! Vol. 44, No. 26 ! 10 September 2005
within the 8–12 !m atmospheric window region.23–28
To infer cloud optical thickness or effective particle
size, information is required on the bulk scattering
properties of these clouds. The single-scattering parameters of ice particles used in the previous studies
are generally based on either those of equivalent ice
spheres27,28 or those of pristine hexagonal ice columns and plates.2,3,19,21
Various particle morphologies including complex
bullet rosettes and aggregates have been observed
frequently in cirrus clouds.29 To account for these ice
particle habits in both IR radiative transfer computations and remote sensing applications, morecomprehensive data sets of the single-scattering
properties of ice particles are required. This study
reports on the development of a comprehensive set of
the scattering and absorption properties for seven
nonspherical ice particle shapes, or habits, in the
near-IR through far-IR spectral region from 3 to
100 !m. The specific ice particle shapes include hexagonal plates, hexagonal solid and hollow columns,
aggregates, three-dimensional (3D) bullet rosettes,
spheroids, and droxtals. The present scattering
computations are carried out for particle maximum
dimensions ranging from 2 to 10,000 !m. The singlescattering properties of the aforementioned ice par-
ticles are computed from a composite method2,3 based
on the finite-difference time-domain (FDTD) technique,12,14,30 an improved geometrical-optics method
(IGOM),17 and the Lorenz–Mie solutions for equivalent spheres. For the single-scattering properties of
spheroids, we use the rigorous T-matrix code developed by Mishchenko and Travis31 for small and moderate size parameters. The size parameter is specified
as x " #D!$, where D is the particle maximum dimension and ! is the wavelength.
The rest of this paper is organized as follows. In
Section 2 we provide the details on the present singlescattering calculations. Results for individual ice particles are provided in Section 3. In Section 4 we
discuss the bulk scattering properties for midlatitude cirrus clouds, and conclusions are presented
in Section 5.
2. Computation of Single-Scattering Properties for
Individual Ice Particles
The fundamental single-scattering parameters required for radiative transfer computations are the
extinction efficiency, single-scattering albedo, and
scattering phase function. The asymmetry factor,
which is the first-order moment of the phase function,
is also widely used in IR radiative flux computations.
These parameters are included in the present database of the single-scattering properties of ice crystals.
The scattering and absorption properties of a particle are determined by the shape and size of the
particle, the complex refractive index, and the incident wavelength. The ice particle shapes considered
in the present computations are shown in Fig. 1,
including aggregates, solid and hollow columns, spheroids, plates, droxtals, and bullet rosettes. For comparison, we also include ice spheres in the present
scattering computations. In this study, all these nonspherical ice particles are assumed to be randomly
oriented in space. The detailed geometry of the droxtal is defined in Yang et al.20 and Zhang et al.32 In
addition to their dependence on particle orientations
and morphology, the scattering and absorption properties also depend on the particle aspect ratio for a
given habit. The aspect ratio " is defined as the ratio
of the width of a particle to its length. For a hexagonal
column of semiwidth a and length L, the aspect ratio
of the hexagonal column is % " 2a!L. Following Yang
et al.18 and the references cited therein, we assume
the relationship between semiwidth a and length L
for a hexagonal column as a " 0.35L when L
& 100 !m and a " 0.348L0.5 for L ' 100 !m. For a
hollow hexagonal column, the hollow cavity depth d is
assumed to vary randomly between 0 and L!2 with
an average of d! " 0.25L, and the aspect ratio is the
same as that of a column with an identical size. For a
plate, we assume the aspect ratio is 1 (i.e., L " 2a) for
a ( 2 !m and L " 2.4883a0.474 for a ' 5 !m; the
aspect ratio varies linearly with a for 2 !m ( a
( 5 !m. For spheroids, we assume that the aspect
ratio of a spheroid is 0.5 (the ratio of the length of the
short axis to the long or rotationally symmetric axis).
For bullet rosettes, the aspect ratio is defined with
respect to individual bullet branches. We use the relationship a " 1.1552L0.63, where a and L are the
semiwidth and length of an individual bullet branch,
respectively. The procedure for defining an aggregate
has been presented by Yang and Liou.33 Detailed
definitions of the 3D geometry for each habit can be
found in Yang et al.18 and the references therein.
The effective particle size3,34,35 for an individual ice
crystal is defined in this study as follows:
De(L) "
Fig. 1. Ice crystal shapes defined for the present scattering calculations.
3 V(L)
,
2 A(L)
(1)
where L is the maximum dimension of a nonspherical
ice particle and A and V are the projected area and
volume of the particle, respectively. Note that the
effective size defined in Eq. (1) is proportional to the
mean path length of rays in the anomalous diffraction
theory or the so-called effective distance.35 Figure 2
shows a comparison of the De–L relationships for various ice habits. For a nonspherical particle, the effective particle size is always smaller than its maximum
dimension. For a given maximum dimension, a
spherical particle has the largest effective size "De
" L# among all the habits shown in Fig. 1. Given the
same maximum dimension and the assumed aspect
ratio in this study, the habits may be sorted by descending effective particle size as follows: sphere,
droxtal, spheroid, aggregate, solid column, hollow column, bullet rosette, and plate. For small effective
particle sizes, owing to the variations of the aspect
ratio, the ranking is slightly different. For example,
10 September 2005 ! Vol. 44, No. 26 ! APPLIED OPTICS
5513
Fig. 2. Effective size versus maximum dimension for various
habits.
for a maximum dimension of 50 !m, the effective size
ranking is sphere, droxtal, solid column, spheroid,
hollow column, plate, bullet rosette, and aggregate.
As particle sizes increase, the De values for solid and
hollow columns, plates, and bullet rosettes become
much smaller than their maximum dimensions because these habits are less compact in comparison
with spheres.
Figure 3 shows the complex refractive index of ice
compiled by Warren36 for a spectral region from 3 to
100 !m. The circles indicate the wavelengths for
which the optical properties are computed in this
study. Detailed light-scattering computations are
performed at 49 discrete wavelengths that properly
account for the sharp gradients in the refractive index at several spectral bands. The computed results
at these wavelengths form the database. In turn, the
scattering parameters can be developed at a higher
spectral resolution through interpolation if required.
Fig. 3. Complex refractive index of ice. Data are taken from
Warren.36
5514
APPLIED OPTICS ! Vol. 44, No. 26 ! 10 September 2005
At present, there is no single method that can be
used to compute the single-scattering properties for
nonspherical ice particles with arbitrary sizes and
shapes. The FDTD method12,14,30 has been applied to
a variety of habits. However, the application of the
FDTD method to cases in which size parameters are
larger than 20 is computationally impractical because of the requirements of computer CPU time and
memory. When large size parameters are involved,
the ray-tracing method is normally used. At IR wavelengths, a discontinuity exists between the FDTD
and the IGOM solutions at a size parameter of approximately 20. Fu et al.2,3 and Baran et al.37 attributed this discontinuity to the tunneling effect38,39
that is neglected in the ray-tracing approach.
Yang et al.19 proposed the stretched scattering potential method (SSPM) to overcome the discontinuity
at these intermediate size parameters. However, the
SSPM is limited to the hexagonal ice particle habit
and has not been modified for application to morecomplex habits such as bullet rosettes or aggregates.
At present, a rigorous method is unavailable to accurately bridge the gap between small and large size
parameters at the IR wavelengths. Although Mitchell40,41 has developed a modified anomalous diffraction approximation to derive approximate solutions
for the extinction and absorption efficiencies, it does
not provide information regarding the phase function
or asymmetry factor.
Figure 4 shows the extinction efficiencies, absorption efficiencies, and asymmetry factors for hollow
columns at a wavelength of 15 !m. The particle sizes
are specified in terms of their maximum dimension.
When the maximum dimension of a hollow column
increases to more than 1000 !m, the singlescattering properties derived from the different
methods converge. From Fig. 4, it is evident that the
Fig. 4. Single-scattering properties of hollow columns from the
composite method based on the FDTD, IGOM, and Lorenz–Mie
solutions "$ " 15 !m#.
equivalent-spherical solution overestimates the extinction and absorption efficiencies, whereas the
IGOM underestimates the extinction and absorption
efficiencies.
To resolve the issue regarding the discontinuity
between the FDTD and the IGOM results at the IR
wavelengths, Fu et al.2 developed a composite method
that is based on a linear combination of the
equivalent-spherical solution and the IGOM nonspherical solution for moderate and large size parameters. The weighting coefficients used to combine the
two solutions are properly selected so that a smooth
transition from the FDTD solution to the composite
solution is achieved (see Appendix A). In terms of
methodology, the composite method is similar to that
proposed by Liou et al.1 in the sense that different
scattering computational methods are applied to different size parameter regions. Note that Liou et al.1
directly combined the FDTD and IGOM results for
small and large size parameters, respectively. Their
approach is approximately applicable at short wavelengths. For the present implementation of the composite method, the composite results combine the
results from the FDTD, IGOM, and Lorenz–Mie
methods, as was done by Fu et al.2,3 To apply the
Lorenz–Mie theory, nonspherical particles are usually converted into a sphere with equivalent volume,
equivalent projected area, or equivalent ratio of volume to projected area. For the Lorenz–Mie solution
involved in the present implementation of the composite method, the volume-based equivalence is used
for the composite solutions for hollow columns and
plates, whereas the equivalence based on volume-toprojected-area ratio is applied to other habits. We use
the FDTD technique for ice particles (except for spheroids) with size parameters less than 20; the composite results are used for larger size parameters. As for
spheroids, we employ the T-matrix method31 to compute the single-scattering properties for small and
moderate size parameters with an aspect ratio of 2.
The composite method is applied to compute the extinction efficiencies, absorption efficiencies, and
asymmetry factors for seven nonspherical particles
with intermediate to large size parameters. The solid
curves in Fig. 4 represent the results from the composite method for hollow columns at a wavelength of
15 !m. Note that the transition from the FDTD solution to the composite results is continuous. At
present, there is no exact computationally efficient
method available to test or validate the accuracy of
the composite results for complex particles (e.g., aggregates and bullet rosettes) with size parameters
between $25 and 60.
A similar approach is used for the derivation of the
scattering phase function. However, the composite
method is not applicable to the phase function computation. The relative angular distribution of the
scattered energy (i.e., the normalized phase function)
computed from IGOM is reasonably accurate42 for
large particles at the IR wavelengths because strong
absorption is involved and the phase function is quite
featureless. For this reason, one should interpret the
phase functions computed from the IGOM as approximate solutions for large size parameters. For small
size parameters (size parameters smaller than 20),
the phase functions from FDTD are employed. For
the present study, the scattering phase functions for
each habit are computed at 498 scattering angles
from 0° to 180°.
3. Results
Figure 5 shows the extinction efficiency, absorption
efficiency, and asymmetry factor as functions of
wavelength and habit for a case where the particle
maximum dimension is fixed at 10 !m. For comparison, results are included for spherical ice particles.
For clarity in Figs. 5– 8, the results for eight ice particle habits are displayed in two columns. The panels
in the left columns show the results for aggregates,
solid columns, spheres, and bullet rosettes, whereas
the panels in the right columns show results for droxtals, hollow columns, plates, and spheroids. In Fig. 5
the variation of the single-scattering properties with
wavelength depends on the refractive index shown in
Fig. 3. As such, the overall variation of the singlescattering properties versus wavelength is similar for
the different habits.
Fig. 5. Variations of extinction efficiency, absorption efficiency,
and asymmetry factor for various habits for a maximum particle
dimension of 10 !m.
10 September 2005 ! Vol. 44, No. 26 ! APPLIED OPTICS
5515
Fig. 6. Variations with wavelength of extinction efficiency, absorption efficiency, and asymmetry factors for various habits for a
fixed effective size of 10 !m.
For each habit, there are several maxima in the
extinction and absorption efficiencies that are associated primarily with the spectral variation of the
refractive index of ice. In particular, the absorption
efficiency has a strong correlation with the imaginary
part of the refractive index (see Fig. 3). There is a
general tendency for the asymmetry factor to decrease with increasing wavelength, which is due to
the fact that the size parameter of a particle decreases with increasing wavelength for a given particle size, leading to a decreased amount of forward
scattering in the phase function. Different habits
have substantially different extinction and absorption efficiencies at a given wavelength due to the
differences in particle volume, effective size, and
shape. With a fixed maximum dimension of 10 !m,
the spherical particle has the largest volume and
effective particle size whereas the corresponding parameters for the aggregate are the smallest (see Fig.
2) by comparison. More generally, the extinction and
absorption efficiencies of the spherical particles are
larger than those for the other habits at most wavelengths, whereas the extinction and absorption efficiencies of the aggregate are the smallest.
5516
APPLIED OPTICS ! Vol. 44, No. 26 ! 10 September 2005
Fig. 7. Variations of extinction efficiency, absorption efficiency,
and asymmetry factor as functions of maximum dimension for
various habits at a fixed wavelength of 8 !m.
Figure 6 shows the extinction efficiency, absorption efficiency, and asymmetry factor as functions of
wavelength for eight particle shapes with a fixed
particle size of 10 !m that is specified in terms of
the effective particle size rather than the particle
maximum dimension as in the previous discussion.
The variation of the extinction and absorption efficiencies for the different habits are more similar for
the case with a given effective particle size than for
the case with a given ice particle maximum dimension, as is evident from a comparison of Figs. 5 and
6. However, at wavelengths with strong absorption,
such as at wavelengths of 3.2, 12.2, and 46.7 !m,
significant differences are still noted in the extinction and absorption efficiencies between the different habits. Part of these differences may be
attributed to the tunneling effect discussed by Baran et al.13 and Mitchell et al.39 The latter authors
also discussed the contribution of photon tunneling
to extinction and absorption efficiencies for nonspherical particles. The rank in absorption efficiency at 12.2 !m in Fig. 6 is sphere, droxtal,
spheroid, solid column, plate, hollow column, bullet
rosette, and aggregate.
Figure 7 shows the variation of the singlescattering properties for various habits with parti-
cle maximum dimension at a wavelength of 8 !m.
The single-scattering properties increase with maximum dimension up to sizes of approximately
20 !m for extinction efficiency and 50 !m for absorption efficiency and asymmetry factor. The extinction efficiencies oscillate around a value of 2 as
the particle size increases further. There are several resonance maxima in the extinction efficiencies, but the locations and magnitude of the
extinction and absorption maxima are different for
each habit. The single-scattering properties of these
ice particles converge to the asymptotic limits given
by the geometrical-optics solution when the maximum dimension is large. For example, the extinction efficiencies for very large particles "D
) 1000 !m# converge to 2, and the absorption coefficients converge to 0.94, which are the asymptotic
values computed from the geometrical-optics
method. The asymmetry factor generally increases
with particle size. From Fig. 7 it is evident that,
given the same particle maximum dimension,
spherical particles have the largest absorption at
this wavelength among all the habits.
Figure 8 is similar to Fig. 7, except that the optical
properties are defined as functions of effective parti-
cle size rather than maximum dimension. From a
comparison of the results in Figs. 7 and 8, the differences between the extinction (and also absorption)
efficiencies for various habits are smaller if the effective particle size is used, particularly for the cases
with small values of the effective particle size. Although the overall variation of the single-scattering
properties with the effective particle size is similar
between these habits, there still are significant differences in the resonance regions. The ranking in
absorption efficiency for an effective particle size of
40 !m in Fig. 8 is sphere, droxtal, spheroid, solid
column, hollow column, bullet rosette, aggregate, and
plate. The extinction and absorption efficiencies for
spherical particles generally are larger than those of
the other habits. This seems to indicate that the assumption of spherical ice particles in the near-IR
through far-IR region is generally not optimal for
radiative transfer computations.
Figure 9 shows the contours of the extinction efficiency, absorption efficiency, and asymmetry factor
as functions of wavelength and maximum dimension
for three habits: solid columns, droxtals, and aggregates. The extinction and absorption efficiencies are
dependent on the incident wavelength and the sizes
and shapes of these ice particles. The absorption efficiency is strongly related to the imaginary part of
the refractive index of ice. In general, the asymmetry
factor increases with particle size and decreases with
wavelength.
Figure 10 shows the scattering phase functions at
a wavelength of 10 !m for three particle habits (aggregate, solid column, and bullet rosette) having a
maximum dimension of 50 !m (top panel) and an
effective size of 50 !m (bottom panel). For a fixed
maximum dimension, differences are noted between
bullet rosettes and the other two habits because of
different volumes of these particles for a given maximum dimension and therefore a different effective
particle size. For habits of a fixed effective particle
size, the scattering phase functions are quite similar.
Compared with those computed at the visible wavelengths,9,18,33 phase functions in the IR region are
essentially featureless in the side-scattering and
backscattering directions. This behavior can be attributed to substantial absorption within the ice particles at the IR wavelengths.
4. Bulk Optical Properties of Cirrus Clouds and
Parameterization
Fig. 8. Extinction efficiency, absorption efficiency, and asymmetry factor as functions of effective size for various habits at a fixed
wavelength of 8 !m.
In this section the single-scattering properties for individual particles, discussed in Section 3, are integrated over particle size distributions that are
representative of mid-latitude cirrus clouds. The
mean (or bulk) optical properties for a given particle
size distribution n"L# and habit distribution [i.e., a
fractional percentage of each ice particle habit fl"L#]
are determined as follows:
10 September 2005 ! Vol. 44, No. 26 ! APPLIED OPTICS
5517
Fig. 9. Contours of extinction efficiency, absorption efficiency, and asymmetry factor as functions of wave number and the maximum
dimension for solid columns [panels in column (a)], droxtals [panels in column (b)], and aggregates [panels in column (c)].
'
%Qe& "
Lmax
() f (L)Q (L)A (L)*n(L)dL
N
'
%Qa& "
Lmax
'
%g& "
Lmin
'
%*& "
Fig. 10. Phase functions for various habits with the same maximum dimension of 50 !m (top panel) and with the same effective
size (bottom panel) of 50 !m at a wavelength of 10 !m.
5518
APPLIED OPTICS ! Vol. 44, No. 26 ! 10 September 2005
Lmin
Lmax
Lmin
i
i
ai
i
() f (L)A (L)*n(L)dL
,
(3)
N
i
i"1
i
() f (L)g (L)Q (L)A (L)*n(L)dL
N
i
i"1
Lmin
'
'
(2)
N
Lmax
Lmax
,
() f (L)Q (L)A (L)*n(L)dL
Lmin
'
i
i"1
Lmax
Lmax
i
N
i"1
Lmin
ei
() f (L)A (L)*n(L)dL
Lmax
Lmin
'
i
i"1
Lmin
i
si
i
() f (L)Q (L)A (L)*n(L)dL
,
(4)
N
i"1
i
si
i
() f (L)Q (L)A (L)*n(L)dL
N
i"1
i
si
i
() f (L)Q (L)A (L)*n(L)dL
N
i"1
i
ei
"1+
%Qa&
%Qe&
,
i
(5)
Fig. 11. Bulk extinction efficiency, absorption efficiency, and asymmetry factor calculated for the 21 size distributions observed for
mid-latitude cirrus clouds (circles). The solid curves are the parameterizations based on Eqs. (7)–(9).
where %Qe&, %Qa&, %g&, and %*& are the mean extinction
efficiency, mean absorption efficiency, mean asymmetry factor, and mean single-scattering albedo, respectively; and N is the number of habits in an ice cloud.
Qs"L# " Qe"L# + Qa"L# is the scattering efficiency; and
Qe"L# and Qa"L# are the extinction and absorption
efficiencies, respectively, calculated for individual ice
particles whose maximum dimensions are denoted by
L. In Eqs. (2)–(5), A is the particle projected area, and
Lmin and Lmax are the minimum and maximum sizes
in the size distribution.
Following Foot,43 Francis et al.,44 and Fu,45 the
effective particle size, defined in Eq. (1) for an individual particle, can be extended for an ice cloud with
a given size distribution and fractional habit amount
(fi, where i is the habit) as follows:
3
De "
2
'
'
Lmax
Lmin
Lmax
Lmin
() f (L)V (L)*n(L)dL
N
i"1
i
i
() f (L)A (L)*n(L)dL
.
(6)
N
i"1
i
i
N
fi " 1 at
The habit fraction fi is defined such that )l"1
each particle size L, where N is the total number of
habits.
To represent mid-latitude cirrus clouds, a set of 21
size distributions used in Fu45 and Fu et al.2 are used
to develop the bulk single-scattering properties.
These size distributions were measured in a variety
of mid-latitude cirrus clouds during several field campaigns. The sources of the data sets for these size
distributions are explained in Fu45 and Fu et al.2
As pointed out by King et al.,46 insufficient information exists concerning ice habits and their typical
percentages for any given size distribution. On the
basis of available in situ observations, their study
assumed that the mid-latitude cirrus clouds consist of
50% bullet rosettes, 25% hollow columns, and 25%
plates when the maximum dimension of an ice particle is smaller than 70 !m. For larger particles, they
assumed that bullet rosettes and aggregates dominate the particle size distribution, where the particles
are composed of 30% aggregates, 30% bullet rosettes,
20% hollow columns, and 20% plates. The assumed
ice crystal habit percentages have also been discussed in Baum et al.47 We use the same habit mixture to derive the mean single-scattering properties
and the parameterizations of mid-latitude cirrus
clouds. The shape and aspect ratio of individual crystals in an aggregate is important to the scattering–
absorption properties of the particle. It is not known
at present whether there is a more realistic approximation for such a complex particle than using solid
columns as employed in the previous study.
Figure 11 shows the mean extinction efficiencies,
absorption efficiencies, and asymmetry factors as
functions of effective particle size for four wave10 September 2005 ! Vol. 44, No. 26 ! APPLIED OPTICS
5519
Table 1. Fitting Coefficients in Eqs. (7)–(9) for the Bulk Single-Scattering Properties of Mid-Latitude Cirrus Clouds
$ "!m#
-1
-2
-3
.0
.1
.2
.3
/0
/1
/2
/3
3.08
3.20
3.33
3.50
3.80
4.10
4.30
4.50
4.75
5.00
5.30
5.60
6.00
6.40
6.80
7.20
7.70
8.00
8.50
9.00
9.50
10.00
10.50
10.80
11.20
11.80
12.20
13.00
14.00
15.00
16.20
17.50
18.80
20.00
21.50
23.00
26.00
30.00
35.00
40.00
46.70
52.00
55.00
57.20
60.00
65.00
75.00
88.00
99.99
23.33
25.53
9.46
37.59
31.26
10.68
20.33
21.79
49.97
60.09
45.07
57.53
97.06
75.49
99.72
95.59
99.29
99.72
99.00
99.77
94.29
88.50
99.04
91.55
7.39
87.80
85.22
99.37
98.47
99.86
99.18
98.89
99.32
88.31
47.62
30.71
31.47
29.73
52.95
98.78
99.19
99.88
92.68
54.06
84.26
99.57
54.56
48.06
37.09
10.85
9.31
1.94
14.56
10.62
2.15
6.88
7.25
19.35
23.60
18.15
24.60
44.73
33.00
44.46
42.37
44.63
44.72
44.51
44.99
42.56
39.59
52.15
57.10
11.92
46.59
43.54
47.04
42.51
40.78
39.70
38.33
36.59
31.76
13.38
4.86
0.06
0.17
26.32
55.69
43.76
34.46
24.16
7.01
25.10
27.83
0.10
0.07
0.08
0.95
10.35
12.39
6.22
22.28
19.56
24.08
27.17
45.48
62.69
59.14
88.00
144.00
121.86
168.53
184.14
234.49
257.19
358.12
472.22
611.35
936.09
1250.54
680.44
14.14
193.12
149.15
155.60
193.18
260.59
392.39
492.45
707.69
730.02
677.86
761.98
1194.78
1597.60
1999.19
1728.50
859.98
1300.18
1559.57
1545.60
1878.09
1841.84
1884.74
1998.92
1994.25
0.834
0.832
0.904
1.025
1.050
1.071
1.057
1.034
1.056
1.064
1.070
1.046
1.018
1.012
1.028
1.019
1.046
1.056
1.064
1.072
1.066
1.111
1.073
1.033
0.953
0.870
0.855
0.838
0.858
0.919
0.960
0.965
0.945
0.946
0.953
0.965
0.980
0.982
1.060
1.028
0.859
0.887
0.845
0.865
0.880
0.864
0.840
0.881
0.920
18.58
41.95
60.88
5.93
2.96
7.70
8.10
11.99
4.00
5.68
5.90
3.84
17.02
14.44
6.07
6.56
14.52
8.68
6.30
7.29
4.71
5.45
9.97
27.33
67.55
83.12
81.19
81.86
92.50
92.00
91.32
91.98
72.08
60.81
25.87
13.00
17.12
30.86
65.56
89.13
98.72
94.55
97.81
69.35
94.93
99.35
43.21
13.24
8.51
15.16
37.19
62.99
27.76
61.78
54.25
39.98
32.21
38.82
53.39
48.59
29.60
28.85
27.00
20.79
21.93
33.89
31.90
35.08
35.77
32.79
36.94
28.43
38.10
70.43
77.94
73.30
71.83
83.58
89.95
94.27
98.24
77.54
68.44
56.40
65.21
68.37
58.80
82.57
94.47
81.39
74.79
69.00
45.75
70.12
70.72
15.27
5.67
15.65
6.33
1.41
64.88
19.02
51.09
139.20
71.11
82.69
15.15
132.49
104.57
11.76
78.68
74.66
11.08
27.95
124.18
70.58
66.37
82.92
45.55
23.67
41.74
97.29
131.93
95.58
87.29
111.61
229.23
476.70
891.45
1382.35
1887.23
1901.77
1765.68
1426.66
1688.58
1933.24
1889.06
1092.92
531.13
993.58
1597.60
1910.65
1923.60
1668.81
1917.51
1770.26
1883.37
0.916
0.940
0.970
0.987
0.982
0.995
0.995
0.995
0.987
0.987
0.987
0.995
0.985
0.998
0.998
0.985
0.995
0.995
0.995
0.995
0.987
0.995
0.991
0.996
0.988
0.975
0.960
0.958
0.981
0.985
0.994
0.988
0.995
0.987
0.990
0.979
0.982
0.973
0.985
0.972
0.871
0.901
0.933
0.919
0.914
0.920
0.911
0.886
0.917
83.92
72.16
17.78
16.31
29.66
44.97
28.33
42.08
20.60
21.24
24.34
53.28
29.00
54.43
54.56
29.53
44.53
45.80
50.22
45.52
37.01
76.35
91.30
39.97
37.72
24.01
7.84
8.84
41.54
23.56
35.96
15.35
13.81
13.12
17.87
16.59
43.75
90.60
5.50
6.00
80.12
31.85
4.24
16.82
15.74
6.37
0.45
10.69
72.84
91.93
79.20
21.47
23.16
38.26
54.27
35.39
48.19
26.14
27.13
30.22
59.95
32.20
60.00
60.46
33.67
50.83
52.65
57.69
52.11
41.58
81.21
93.51
42.69
42.24
29.11
11.54
13.26
50.97
31.79
45.71
23.94
22.53
21.77
29.40
26.83
55.52
97.75
8.73
10.66
84.14
36.35
12.17
22.22
19.62
13.58
8.56
19.52
95.96
27.88
27.08
5.55
4.29
19.17
33.96
6.95
28.41
2.69
2.30
2.13
30.08
18.62
36.88
37.20
20.00
28.78
28.74
31.32
29.00
28.94
83.90
121.69
53.09
47.94
25.23
3.11
2.81
56.52
24.97
57.28
5.24
5.77
8.19
7.34
16.42
136.15
409.89
49.00
68.89
1024.19
490.39
99.76
271.70
278.05
163.96
49.79
146.24
665.96
lengths computed for the 21 size distributions. The
bulk scattering properties of ice clouds are shown to
be strongly dependent on the effective particle size at
each wavelength.
A parameterization of these bulk single-scattering
properties is developed based on these results. The
single-scattering properties are parameterized as the
function of effective particle size at each wavelength
as follows:
5520
APPLIED OPTICS ! Vol. 44, No. 26 ! 10 September 2005
%Qe& "
%Qa& "
%g& "
2 , -1De+1
1 , -2De+1 , -3De+2
.0 , .1De+1
1 , .2De+1 , .3De+2
/0 , /1De+1
1 , /2De+1 , /3De+2
,
(7)
,
(8)
,
(9)
dependence on the effective particle size in the IR
region, particularly at small sizes, and are also sensitive to the wavelength. At the near-IR wavelengths
(e.g., 3.08 !m), the extinction efficiency varies
smoothly as a function of effective particle size. At
longer wavelengths, the general feature to note is
that the mean extinction efficiency increases monotonically with size to an asymptotic value of 2. The
3.08 and 12.2 !m wavelengths are located in strong
absorption bands of ice (see the lower panel of Fig. 3),
whereas the 8 and 20 !m wavelengths are located at
the bands with relatively weaker absorption. The
variation of absorption efficiency with effective particle size is different at strongly versus weakly
absorbing wavelengths. At strongly absorbing wavelengths, the absorption efficiency decreases slightly
with increasing effective particle size, whereas the
absorption efficiency increases with increasing effective particle size at weakly absorbing wavelengths.
This is a result of the combination of the particle size
distribution with the absorption properties of the various particle habits at these wavelengths. The mean
asymmetry factor increases with effective particle
size at each wavelength.
The parameterization results can be easily extended for applications involving high-spectralresolution studies through an interpolation of the
fitted results based on the results for the 49 wavelengths. The symbols in Fig. 12 show the parameterized single-scattering properties as a function of
wavelength for three effective particle sizes. Also
shown in Fig. 12 are the results obtained through
interpolation using a spline fitting technique at a
spectral resolution of 0.01 !m. The interpolation does
not incur noticeable errors.
5. Summary
Fig. 12. Bulk single-scattering properties as functions of wavelength for three effective sizes. The symbols show the composite
values at 49 discrete wavelengths; the solid curves are the interpolated results with a high spectral resolution of 0.01 !m.
where -i, .i, and /i "i " 0, 1, 2, 3# are fitting coefficients that are functions of wavelength. These coefficients are determined using the Monte Carlo
technique. The parameterization scheme provided in
Eqs. (7)–(9) has some advantages as compared with
the conventional parameterization scheme based on
polynomials. For example, when De is large, the parameterized extinction efficiency given by Eq. (7) approaches 2, the asymptotic value for large particles,
whereas it approaches zero for small particles. This
parameterization approach has physically correct
limits that are expected from scattering theory. Table
1 lists these fitting coefficients for the 49 wavelengths
used in the database. As an example, the parameterization results for the single-scattering properties at
the four wavelengths are also shown in Fig. 11 (solid
curves).
It is evident from Fig. 11 that the single-scattering
properties for each size distribution have a strong
The single-scattering properties of ice particles in the
near-infrared through far-infrared wavelength regime are computed from a composite method that is
based on the FDTD technique, the T-matrix method,
an IGOM, and the Lorenz–Mie theory for equivalent
ice spheres. Seven nonspherical ice crystal habits (aggregates, hexagonal hollow and solid columns, plates,
3D bullet rosettes, spheroids, and droxtals) are considered in the present study. A database of the singlescattering properties for each of these ice particle
habits has been developed at 49 discrete wavelengths
between 3 and 100 !m and for particle sizes ranging
from 2 to 10,000 !m in terms of the particle maximum dimension. The dependence of the singlescattering properties on the particle maximum
dimension and effective particle size are also discussed. Finally, a parameterization of the bulk optical properties is developed for mid-latitude cirrus
clouds based on 21 particle size distributions obtained from various field campaigns.
Appendix A: Composite Method
In the present study, the derivation of the ice particle
scattering property database employs a composite
method developed by Fu et al.2,3 For completeness, we
10 September 2005 ! Vol. 44, No. 26 ! APPLIED OPTICS
5521
briefly present the theoretical framework of this
method. Basically, the composite method combines
results for the scattering properties (e.g., the extinction efficiency, absorption efficiency, and asymmetry
factor) computed from the IGOM, Lorenz–Mie, and
FDTD methods (see Fig. 4). As an example, the composite result for absorption efficiency at wavelength !
is as follows:
Qa(L) "
+
QaFDTD(L)
(L & 20$!#)
,
C1QaMie(L) , C2QaIGOM(L) (L ' 20$!#)
(A1)
where Qa"L#, QaFDTD"L#, QaMie"L#, and QaIGOM"L# are
absorption coefficients computed from the composite
method, FDTD, the Lorenz–Mie theory, and IGOM,
respectively. The weighting coefficients C1 and C2 in
Eq. (A1) are determined so that a smooth transition is
achieved from the FDTD solution to the composite
solution, i.e.,
QaFDTD(Lc) " C1QaMie(Lc) , C2QaIGOM(Lc),
(A2)
where Lc is the largest maximum dimension derived
from the FDTD method. The coefficients C1 and C2
are dependent on both wavelength and ice particle
habit. The composite method results for the hollow
column habit at a wavelength of 15 !m is shown in
Fig. 4. A similar approach is used to compute extinction efficiencies and asymmetry factors at each wavelength for each ice particle habit.
This research was supported by the National Science Foundation CAREER Award research grant
(ATM-0239605), a research grant from NASA Radiation Sciences Program (NAG-1-02002) managed by
Hal Maring (and previously by Donald Anderson),
and a subcontract (G066010) from the University of
Wisconsin-Madison.
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